The Annals of Probability

Stochastic equations, flows and measure-valued processes

Donald A. Dawson and Zenghu Li

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We first prove some general results on pathwise uniqueness, comparison property and existence of nonnegative strong solutions of stochastic equations driven by white noises and Poisson random measures. The results are then used to prove the strong existence of two classes of stochastic flows associated with coalescents with multiple collisions, that is, generalized Fleming–Viot flows and flows of continuous-state branching processes with immigration. One of them unifies the different treatments of three kinds of flows in Bertoin and Le Gall [Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 307–333]. Two scaling limit theorems for the generalized Fleming–Viot flows are proved, which lead to sub-critical branching immigration superprocesses. From those theorems we derive easily a generalization of the limit theorem for finite point motions of the flows in Bertoin and Le Gall [Illinois J. Math. 50 (2006) 147–181].

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Ann. Probab., Volume 40, Number 2 (2012), 813-857.

First available in Project Euclid: 26 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G09: Exchangeability 60J68: Superprocesses
Secondary: 60J25: Continuous-time Markov processes on general state spaces 92D25: Population dynamics (general)

Stochastic equation strong solution stochastic flow coalescent generalized Fleming–Viot process continuous-state branching process immigration superprocess


Dawson, Donald A.; Li, Zenghu. Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 (2012), no. 2, 813--857. doi:10.1214/10-AOP629.

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